Proof of the set of compatible observable have a bound Sakurai states that

We can obviously generalize our considerations to a situation where there are several mutually compatible observables, namely,
$$[A,B]=[B,C]=[A,C]=\cdots=0$$
Assume that we have found a maximal set of commuting observables, that is; we can't add any more observables to our list without violating the above.

I don't understand, How are you sure that you always found such a bound? Is there any proof or reasoning that can be given?
 A: The statement is correct but physically a bit misplaced as it stands (I like Sakurai's book however).
The existence of a maximal set of commuting observables, strictly referring to  the statement in the question, arises immediately from Zorn's lemma. However it is easy to prove that a maximal such set contains infinite many operators even if the Hilbert space is finite dimensional.
To be precise, in the general case of unbounded observables,  "commuting observables" should understood in the sense that the spectral measures of the observables commute, even if the observables themselves do not, in view of problems with different domains.
Nevertheless, that is not the physically correct definition of maximal set of commuting observables, because it is not restrictive enough, since it does not consider the fact that functions of a set of compatible observables commute with these observables! This is the reason why I wrote that the set always contains infinite elements. (Maybe it is later stated in the book, now I cannot check and I am referring to the portion of text presented above).
The physically correct definition, producing in some cases finite sets of observables is the following one.
A maximal set of mutually compatible observables is a set of mutually compatible observables such that every observable that commutes with the elements of the set is a function of them.
In this case there are important concrete  examples of maximal sets of mutually compatible observables and and an explicit  "existence proof" is the best a physicist should wish. Consider a particle with spin $1/2$, hence described in $L^2(\mathbb{R},dx)\otimes \mathbb{C}^2$. A maximal set of mutually compatible observables is made of the position operator $X$ and the spin operator $S_z$ along $z$. (The proof of this fact is not trivial.)
It is not difficult to prove that if $A_1,\ldots, A_n$ is a maximal  set of mutually compatible observables each with point spectrum, then a simultaneous non-destructive measurement of them produces a pure state as post measurement state. Measurements of maximal  set of mutually compatible observables are therefore used to prepare pure states.
